Only 1% Can Solve This "Impossible" System

In this step by step math tutorial, we solve the nonlinear simultaneous system x²+y⁴ = 20 and x⁴+y² = 20 — a beautifully symmetric pair of equations that rewards careful structural observation far more than brute force algebra. The two equations are symmetric in x and y in the sense that swapping x and y transforms one equation into the other, and that observation is the key that unlocks the entire solution. We subtract one equation from the other, factor the result, analyse all cases systematically, and find every real solution pair completely and rigorously. Every step is explained clearly from start to finish. This is exactly the kind of problem where students who reach for elimination or substitution immediately will spend a very long time getting nowhere, while students who pause and notice the symmetry will solve it cleanly in minutes. What you will learn: How subtracting one equation from the other reveals a powerful factorization. How to analyse all cases that arise from the factored equation systematically. How to find all real solution pairs completely and rigorously. How to verify every solution pair in both original equations. This type of symmetric nonlinear system is a favourite in math olympiad competitions, IB Mathematics Higher Level, A-Level Further Mathematics, college entrance examinations, and university precalculus courses. Recognising and exploiting symmetry is one of the most powerful and elegant problem solving tools in all of mathematics. If you found this helpful, give it a thumbs up, share it with a fellow math lover, and subscribe for weekly videos on algebra, calculus, number theory, and olympiad problem solving. Hit the notification bell so you never miss an upload. Don’t forget to like 👍, subscribe    / @nonsomaths  , and hit the notification bell for more math tips and tricks! #maths #algebra #matholympiad